Noetherʹs problem for groups of order 32

Let K be any field, G be a finite group. Let G act on the rational function field by K-automorphisms defined by h xg=xhg for any g,h G. Denote by the fixed field. Noetherʹs problem asks, under what situations, the fixed field K(G) will be rational (= purely transcendental) over K. Theorem Let G be a finite group of order 32 with exponent e. If charK=2 or K is any field containing a primitive eth root of unity, then K(G) is rational over K.